# example of four exponentials conjecture

Taking $x_{1}=i\pi$, $x_{2}=i\pi\sqrt{2}$, $y_{1}=1$, $y_{2}=\sqrt{2}$, we see that this conjecture implies that one of $e^{i\pi}$, $e^{i\pi\sqrt{2}}$, or $e^{i2\pi}$ is transcendental. Since the first is $-1$ and the last is $1$, the conjecture states that second must be transcendental, that is, $e^{i\pi\sqrt{2}}$ is (conjecturally) transcendental.

In this particular case, the result is known already, so the conjecture is verified. Using Gelfond’s theorem, take $\alpha=e^{i\pi}$ and $\beta=\sqrt{2}$ and it follows that $\alpha^{\beta}$ is transcendental.

Title example of four exponentials conjecture ExampleOfFourExponentialsConjecture 2013-03-22 14:09:09 2013-03-22 14:09:09 archibal (4430) archibal (4430) 6 archibal (4430) Example msc 11J81